Time-independent Hamiltonian Systems Research Articles

Reactive islands for three degrees-of-freedom Hamiltonian systems

We develop the geometrical, analytical, and computational framework for reactive island theory for three degrees-of-freedom time-independent Hamiltonian systems. In this setting, the dynamics occurs in a 5-dimensional energy surface in phase space and is governed by four-dimensional stable and unstable manifolds of a three-dimensional normally hyperbolic invariant sphere. The stable and unstable manifolds have the geometrical structure of spherinders and we provide the means to investigate the ways in which these spherinders and their intersections determine the dynamical evolution of trajectories. This geometrical picture is realized through the computational technique of Lagrangian descriptors. In a set of trajectories, Lagrangian descriptors allow us to identify the ones closest to a stable or unstable manifold. Using an approximation of the manifold on a surface of section we are able to calculate the flux between two regions of the energy surface.

Single-Hessian thawed Gaussian approximation.

To alleviate the computational cost associated with on-the-fly ab initio semiclassical calculations of molecular spectra, we propose the single-Hessian thawed Gaussian approximation in which the Hessian of the potential energy at all points along an anharmonic classical trajectory is approximated by a constant matrix. The spectra obtained with this approximation are compared with the exact quantum spectra of a one-dimensional Morse potential and with the experimental spectra of ammonia and quinquethiophene. In all cases, the single-Hessian version performs almost as well as the much more expensive on-the-fly ab initio thawed Gaussian approximation and significantly better than the global harmonic schemes. Remarkably, unlike the thawed Gaussian approximation, the proposed method conserves energy exactly, despite the time dependence of the corresponding effective Hamiltonian, and, in addition, can be mapped to a higher-dimensional time-independent classical Hamiltonian system. We also provide a detailed comparison with several related approximations used for accelerating prefactor calculations in semiclassical simulations.

Complex integrals for 3-dimensional non-hermitian Hamiltonian systems

With a view to obtain further insight into the theoretical understanding of complex potentials, we here investigated the dynamical invariants for time-independent (TID) complex Hamiltonian systems in three dimensions by invoking the elegante extended complex phase space approach (ECPSA) characterized by x=x1+ip4,y=x2+ip5,z=x3+ip6,px=p1+ix4,py=p2+ix5 and pz=p3+ix6. We make use of rationalization method for this purpose and the invariants so obtained are expected to play an important role in studying the non-hermitian Hamiltonian systems.

On the invariant method for the time-dependent non-Hermitian Hamiltonians

We propose a scheme to deal with certain time-dependent non-Hermitian Hamiltonian operators $H(t)$ that generate a real phase in their time-evolution. This involves the use of invariant operators $I_{PH}(t)$ that are pseudo-Hermitian with respect to the time-dependent metric operator, which implies that the dynamics is governed by unitary time evolution. Furthermore, $H(t)$ is not generally quasi-Hermitian and does not define an observable of the system but $I_{PH}(t)$ obeys a quasi-hermiticity transformation as in the completely time-independent Hamiltonian systems case. The harmonic oscillator with a time-dependent frequency under the action of a complex time-dependent linear potential is considered as an illustrative example.

Exact analytical solutions for time-dependent Hermitian Hamiltonian systems from static unobservable non-Hermitian Hamiltonians

We propose a procedure to obtain exact analytical solutions to the time-dependent Schr\"{o}dinger equations involving explicit time-dependent Hermitian Hamitonians from solutions to time-independent non-Hermitian Hamiltonian systems and the time-dependent Dyson relation together with the time-dependent quasi-Hermiticity relation. We illustrate the working of this method for a simple Hermitian Rabi-type model by relating it to a non-Hermitian time-independent system corresponding to the one-site lattice Yang-Lee model.

Universality of fractal dimension on time-independent Hamiltonian systems

This paper summarizes a numerical study of the dependence of the fractal dimension on the energy of certain open Hamiltonian systems, which present different kind of symmetries. Owing to the presence of chaos in these systems, it is not possible to make predictions on the way and the time of escape of the orbits starting inside the potential well. This fact causes the appearance of fractal boundaries in the initial-condition phase space. In order to compute its dimension, we use a simple method based on the perturbed orbits’ behavior. The results show that the fractal dimension function depends on the structure of the potential well, contrary to other properties, such us the probability of escape, which has already been postulated as universal in earlier papers (see for instance [C. Siopis, H.E. Kandrup, G. Contopoulos, R. Dvorak, Universal properties of escape in dynamical systems, Celest. Mech. Dyn. Astr. 65 (57–68) (1997)]), from the study of Hamiltonians with different number of possible exits.

Applications of Hofer's geometry to Hamiltonian dynamics

We prove that for every subset $A$ of a tame symplectic manifold $(W,\omega)$ meeting a semi-positivity condition, the $\pi_1$-sensitive Hofer--Zehnder capacity of $A$ is not greater than four times the stable displacement energy of $A$, $$ c_{HZ}^\circ(A,W)\le 4e (A\times S^1, W\times T^*S^1). $$ This estimate yields almost existence of periodic orbits near stably displaceable energy levels of time-independent Hamiltonian systems. Our main applications are: $\bullet$ The Weinstein conjecture holds true for every stably displaceable hypersurface of contact type in $(W,\omega)$. $\bullet$ The flow describing the motion of a charge on a closed Riemannian manifold subject to a non-vanishing magnetic field and a conservative force field has contractible periodic orbits at almost all sufficiently small energies. The proof of the above energy-capacity inequality combines a curve shortening procedure in Hofer geometry with the following detection mechanism for periodic orbits: If the ray $\{\varphi_F^t \}$, $t \ge 0$, of Hamiltonian diffeomorphisms generated by a compactly supported time-independent Hamiltonian stops to be a minimal geodesic in its homotopy class, then a non-constant contractible periodic orbit must appear

Phase Mixing in Unperturbed and Perturbed Hamiltonian Systems

This paper summarises a numerical investigation of phase mixing in time-independent Hamiltonian systems that admit a coexistence of regular and chaotic phase space regions, allowing also for low amplitude perturbations idealised as periodic driving, friction, and/or white and coloured noise. The evolution of initially localised ensembles of orbits was probed through lower order moments and coarse-grained distribution functions. In the absence of time-dependent perturbations, regular ensembles disperse initially as a power law in time and only exhibit a coarse-grained approach towards an invariant equilibrium over comparatively long times. Chaotic ensembles generally diverge exponentially fast on a time scale related to a typical finite time Lyapunov exponent, but can exhibit complex behaviour if they are impacted by the effects of cantori or the Arnold web. Viewed over somewhat longer times, chaotic ensembles typical converge exponentially towards an invariant or near-invariant equilibrium. This, however, need not correspond to a true equilibrium, which may only be approached over very long time scales. Time-dependent perturbations can dramatically increase the efficiency of phase mixing, both by accelerating the approach towards a near-equilibrium and by facilitating diffusion through cantori or along the Arnold web so as to accelerate the approach towards a true equilibrium. The efficacy of such perturbations typically scales logarithmically in amplitude, but is comparatively insensitive to most other details, a conclusion which reinforces the interpretation that the perturbations act via a resonant coupling.

Chaotic Level Statistics in One-Dimensional Conservative System

Quantum mechanics of one-dimensional time-independent Hamiltonian system whose energy level statistics obeys the Gaussian ensemble is numerically studied by the inverse scattering method. It has been known that the system must be integrable. However, at least numerically, it can be shown that we can construct a potential for the Schrodinger equation that reproduces a finite number of given energy levels of chaotic regime, i.e. given by the random matrix theory (RMT). Contrary to usual classical Hamiltonian systems, there cannot exit the shortest period from the consequence of the semi-classical argument, if the system completely reproduces the chaotic levels of the RMT. If its density of states is unfolded and its average density is independent of energy, the coarse averaged shape of the potential is just like a harmonic oscillator and the potential has peculiar ripples on it. The more energy levels of chaotic regime we take, the more intricate and the finer these ripples become.

Chaos, ergodicity, and the thermodynamics of lower-dimensional time-independent Hamiltonian systems.

This paper uses the assumptions of ergodicity and a microcanonical distribution to compute estimates of the largest Lyapunov exponents in lower-dimensional Hamiltonian systems. That the resulting estimates are in reasonable agreement with the actual values computed numerically corroborates the intuition that chaos in such systems can be understood as arising generically from a parametric instability and that this instability may be modeled by a stochastic-oscillator equation [cf. Casetti, Clementi, and Pettini, Phys. Rev. E 54, 5969 (1996)], linearized perturbations of a chaotic orbit satisfying a harmonic-oscillator equation with a randomly varying frequency.

Exact coherent states in one-dimensional quantum many-body systems with inverse-square interactions

For the models of $N$-body identical harmonic oscillators interacting through potentials of homogeneous degree -2, the unitary operator that transforms a system of time-dependent parameters into that of unit spring constant and unit mass of different timescale is found. If the interactions can be written in terms of the differences between positions of two particles, it is also shown that the Schrodinger equation is invariant under a unitary transformation. These unitary relations can be used not only in finding coherent states from the given stationary states in a system, but also in finding exact wave functions of the Hamiltonian systems of time-dependent parameters from those of time-independent Hamiltonian systems. Both operators are invariant under the exchange of any pair of particles. The transformations are explicitly applied for some of the Calogero-Sutherland models to find exact coherent states.

Phase space transport in noisy Hamiltonian systems.

This paper analyzes the effect of low amplitude friction and noise in accelerating phase space transport in time-independent Hamiltonian systems that exhibit global stochasticity. Numerical experiments reveal that even very weak non-Hamiltonian perturbations can dramatically increase the rate at which an ensemble of orbits penetrates obstructions like cantori or Arnold webs, thus accelerating the approach toward an invariant measure, i.e., a microcanonical population of the accessible phase space region. An investigation of first passage times through cantori leads to three conclusions, namely: (i) that, at least for white noise, the detailed form of the perturbation is unimportant, (ii) that the presence or absence of friction is largely irrelevant, and (iii) that, overall, the amplitude of the response to weak noise scales logarithmically in the amplitude of the noise.

Resonances in a spring-pendulum: algorithms for equivariant singularity theory

A spring-pendulum in resonance is a time-independent Hamiltonian model system for formal reduction to one degree of freedom, where some symmetry (reversibility) is maintained. The reduction is handled by equivariant singularity theory with a distinguished parameter, yielding an integrable approximation of the Poincaré map. This makes a concise description of certain bifurcations possible. The computation of reparametrizations from normal form to the actual system is performed by Gröbner basis techniques.

Phase mixing in time-independent Hamiltonian systems

Everything you ever wanted to know about what has come to be known as ``chaotic mixing:'' This paper describes the evolution of localised ensembles of initial conditions in 2- and 3-D time-independent potentials which admit both regular and chaotic orbits. The coarse-grained approach towards an invariant, or near-invariant, distribution was probed by tracking (1) phase space moments through order 4 and (2) binned reduced distributions f(Z_a,Z_b,t) for a,b=x,y,z,p_x,p_y,p_z, computed at fixed time intervals. For ``unconfined'' chaotic orbits in 2-D systems not stuck near islands by cantori, the moments evolve exponentially: Quantities like the dispersion in p_x, which start small and eventually asymptote towards a larger value, initially grow exponentially in time at a rate comparable to the largest short time Lyapunov exponent. Quantities like | |, that can start large but eventually asymptote towards zero, decrease exponentially. With respect to a discrete L^p norm, reduced distributions f(t) generated from successive decay exponentially towards a near-invariant f_{niv}, although a plot of Df(t)=||f(t)-f_{niv}|| can exhibit considerable structure. Regular ensembles behave very differently, both moments and Df evolving in a fashion better represented by a power law time dependence. ``Confined'' chaotic orbits, initially stuck near regular islands because of cantori, exhibit an intermediate behaviour. The behaviour of ensembles evolved in 3-D potentials is qualitatively similar, except that, in this case, it is relatively likely to find one direction in configuration space which is ``less chaotic'' than the other two, so that quantities like L_{ab} depend more sensitively on which phase space variables one tracks.

Some aspects of Hamiltonian systems and symplectic algorithms

Symplectic algorithms have been shown to be a right formalism for numerical computation of Hamiltonian systems. They are suitable to long time computation and of good qualitative properties. These properties are ensured by the fact that a symplectic algorithm approximating to a time-independent Hamiltonian system can be regarded as a perturbed time-dependent Hamiltonian system of the original one. That is, a solution of a symplectic algorithm is a solution of a certain perturbed time dependent Hamiltonian system evaluated at (time) discrete points. Moreover, a linear symplectic algorithm approximating to a linear time-independent Hamiltonian system can be regarded as a perturbed time-independent Hamiltonian system. So it has all properties that a linear Hamiltonian system has. Based on these results, stochastic webs and chaos appeared in symplectic simulation for Hamiltonian systems are also discussed.

Temporal crossover from classical to quantum behavior: a Markov-chain approach

Previous renormalization group analyses that for chaotic systems near their critical points, the crossover time from classical to quantum behaviors scales with the Planck constant ħ as t max∼ħ -μ. We argue that the same scaling relation also holds for typical two-degrees-of-freedom and time-independent chaotic Hamiltonian systems. Our analysis makes use of a self-similar Markov-chain model which was previously used to qualitatively explain the algebraic decay law in Hamiltonian systems.

Symbolic computation of normal forms for nonlinear Hamiltonian systems

In this paper a method of computing normal forms for time-independent nonlinear Hamiltonian systems is given. A program using the symbolic manipulator MACSYMA is used to do an example.

Dissipative evolution, initial conditions, and information theory.

An information-theoretic approach is used to give a straightforward procedure that allows one to determine whether dissipative behavior can be attained for a given Hamiltonian. This technique stresses the importance of an adequate set of initial conditions in order to obtain dissipative evolution. It is shown that, although dissipative solutions may be compatible with the set of dynamical equations, they may not be realized just because it is impossible to construct the adequate set of initial conditions that would result in dissipative behavior. This is illustrated by reference to the well-known Bateman generalized problem of two interacting harmonic oscillators. We demonstrate the impossibility of obtaining dissipative evolution in linear, time-independent Hamiltonian systems generated with any combination of bilinear products of creation and annihilation operators.

Semiclassical functional methods for arbitrary Hamiltonians

The semiclassical formula for the propagator is derived for arbitrary time-independent Hamiltonian systems with no constraints using Feynman's path-integral approach.